What is the physical dimension/unit of Probability current?

1. The problem statement, all variables and given/known data
Question:
What is the physical dimension of Probability Current for a particle in 1 dimension? (Quantum Mechanics)

2. Relevant equations
Quantum mechanical Probability Current:

3. The attempt at a solution
I know the physical dimension of mass, that is kg. If I know every dimension, I can try some things and I can find the dimension of the observable. But now, I'm stuck. I guess that the wave function has no dimension, because it is very related to probability. But what's the case with $$ \frac{\partial \Psi}{\partial x} $$?

I guess that the wave function has no dimension, because it is very related to probability.

So here we must react also: It is related to probability in the sense that $$\int \psi^*\psi\,d\tau = 1 $$where the integral is over all space. The ##1## is dimensionless: a genuine probability. So guess again !

So here we must react also: It is related to probability in the sense that $$\int \psi^*\psi\,d\tau = 1 $$where the integral is over all space. The ##1## is dimensionless: a genuine probability. So guess again !

This is a nice question to answer. It took me some time but I think:
∫ψ∗ψdτ=1 has no dimensions.
But as you integrate over space, and in this case this is one-dimensional, the dimension of ψ∗ψ is multiplied with [length]. And then ψ has a dimension of [(1/length)^½]?

An unavoidable conclusion, isn't it ? I never worried about the wave function having a dimension (always considered it as dimensionless) and would have liked to keep it that way. But -- unless we are being corrected -- this is what comes out !